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In this paper, we study conformally flat ($\alpha,\beta$)-metrics in the form $F=\alpha(1+\sum_{j=1}^m a_j(\frac{\beta}{\alpha})^j)$ with $m\geq2$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form on a smooth manifold $M$. We prove that if such conformally flat ($\alpha,\beta$)-metric $F$ is of weakly isotropic scalar curvature, then it must has zero scalar curvature. Moreover, if $a_{m-1} a_m\neq0$, then such metric is either locally Minkowskian or Riemannian.
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참고문헌 (16)
참고문헌 신청
H. Akbar-Zadeh / 1988 / Sur les espaces de Finsler a courbures sectionnelles constantes / Acad. Roy. Belg. Bull. Cl. Sci 74(10):281~322
D. Bao / 2000 / An Introduction to Riemann-Finsler Geometry / Springer-Verlag
G. Chen / 2013 / An important class of conformally at weak Einstein Finsler metrics / Internat. J. Math 24(1):15
G. Chen / 2015 / On conformally at (α; β)-metrics with constant ag curvature / Publ. Math. Debrecen 86(3-4):387~400
X. Cheng / 2009 / A class of Finsler metrics with isotropic S-curvature / Israel J. Math 169:317~340
D. Bao / 2000 / An Introduction to Riemann-Finsler Geometry / Springer-Verlag
G. Chen / 2013 / An important class of conformally at weak Einstein Finsler metrics / Internat. J. Math 24(1):15
G. Chen / 2015 / On conformally at (α; β)-metrics with constant ag curvature / Publ. Math. Debrecen 86(3-4):387~400
X. Cheng / 2009 / A class of Finsler metrics with isotropic S-curvature / Israel J. Math 169:317~340
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